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We usually construct $\Bbb R$ from Dedekind cuts. It is surprising to me that $|\mathcal{P}(\Bbb N)|=|\Bbb R|$.

Is it possible to construct $\Bbb R$ directly from $\mathcal{P}(\Bbb N)$ rather than $\Bbb N \to \Bbb Z \to \Bbb Q \to \Bbb R$? In that construction, we associate a real number with a subset of $\Bbb N$ and acquire familiar operations such as addition, subtraction, multiplication, and division.

Isaac Browne
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Akira
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    It's definitely possible to construct a bijection. But a construction which makes the arithmetic on the resulting $\Bbb R$ naturally inherited from the arithmetic on $\Bbb N$ in any way is going to be a real challenge I think. That's one of the real advantages of the standard $\Bbb N\to\Bbb Z\to\Bbb Q\to\Bbb R$ construction. – Arthur Oct 09 '18 at 10:19
  • Thank you so much @Arthur! I got your point. – Akira Oct 09 '18 at 10:23

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This construction is in the right 'ballpark' at least, going from $\mathbb Z$ to $\mathbb R$ without the necessity of constructing $\mathbb Q$.

A function $f: \mathbb Z \to \mathbb Z$ is identified, by its graph, with a subset of $\mathcal{P}(\Bbb Z \times \Bbb Z)$. We can define a class of special functions that respect the addition operation on $\Bbb Z$ is some way, and then, those functions can be used to construct $\Bbb R$.

See the wikipedia article Construction from Z (Eudoxus reals).

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